Related papers: One-parameter Darboux-deformed Fibonacci numbers
A new procedure for the global construction of the Casimir invariants and Darboux canonical form for finite-dimensional Poisson systems is developed. This approach is based on the concept of matrix congruence and can be applied without the…
The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the $N$-fold application of the transformation is also established, and…
A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian as well as the corresponding eigenfunctions are determined…
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…
The law of transformation of affine connection for n-dimensional manifolds as the system of nonlinear equations on local coordinates of manifold is considered. The extension of the Darboux-Lame system of equations to the spaces of constant…
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for…
This paper presents a new approach to the Hamiltonian structure of isomonodromic deformations of a matrix system of ODE's on a torus. An isomonodromic analogue of the $\rmSU(2)$ Calogero-Gaudin system is used for a case study of this…
We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As…
We describe generally deformed Heisenberg algebras in one dimension. The condition for a generalized Leibniz rule is obtained and solved. We analyze conditions under which deformed quantum-mechanical problems have a Fock-space…
Irreducible second-order Darboux transformations are applied to the periodic Schrodinger's operators. It is shown that for the pairs of factorization energies inside of the same forbidden band they can create new non-singular potentials…
Using the Fourier-Laplace transform, we describe the isomonodromy equations for meromorphic connections on the Riemann sphere with unramified irregular singularities as those for connections with a (possibly ramified) irregular singularity…
We study a complex intertwining relation of second order for Schroedinger operators and construct third order symmetry operators for them. A modification of this approach leads to a higher order shape invariance. We analyze with particular…
A general framework for the deformation of the single-mode oscillators is presented and all deformed single-mode oscillators are unified. The extensions of the Aric-Coon, genon, the para-Bose and the para-Fermi oscillators are proposed. The…
In the KdV context we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann-Hilbert problem and yields a new explicit formula for perturbation of the…
q-Deformed harmonic oscillator algebra for real and root of unity values of the deformation parameter is discussed by using an extension of the number concept proposed by Gauss, namely the Q-numbers. A study of the reducibility of the Fock…
The discrete non-commutative Darboux system of equations with self-consistent sources is constructed, utilizing both the vectorial fundamental (binary Darboux) transformation and the method of additional independent variables. Then the…
The Gauss-Manin equations are solved for a class of flat-metrics defined by Novikov algebras, this generalizing a result of Balinskii and Novikov who solved this problem in the case of commutative Novikov algebras (where the algebraic…
We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference…
We study the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^3$, in the low-regularity regime. We introduce a notion of weak solution for $u\in…
Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems", Comm. Math. Phys. 207, pp. 589-620 (1999), we have…